Motives with exceptional Galois groups and the inverse Galois problem

TL;DR

This paper constructs motivic $ ext{ell}$-adic Galois representations with dense images in exceptional groups, providing new solutions to the inverse Galois problem by realizing certain finite simple groups as Galois groups over $ ext{Q}$.

Contribution

It introduces a uniform construction of motivic $ ext{ell}$-adic representations into exceptional groups, answering Serre's question and expanding the known cases of the inverse Galois problem.

Findings

Constructed motivic $ ext{ell}$-adic representations with Zariski dense images in $E_7$, $E_8$, and $G_2$.

Realized $E_8( ext{F}_ ext{ell})$ as Galois groups over $ ext{Q}$ for large primes $ ext{ell}$.

Provided a new approach using the Langlands correspondence for function fields.

Abstract

We construct motivic $\ell$-adic representations of $\GQ$ into exceptional groups of type $E_7,E_8$ and $G_2$ whose image is Zariski dense. This answers a question of Serre. The construction is uniform for these groups and uses the Langlands correspondence for function fields. As an application, we solve new cases of the inverse Galois problem: the finite simple groups $E_{8}(\FF_{\ell})$ are Galois groups over $\QQ$ for large enough primes $\ell$.